The problem was that tab and enter did trigger a refresh on the suggest but due to the timeout-nature hiding the suggest due to blur was before showing it again because of the change to the element. I hope this makes sense.

Ideally this should use ​UI Autocomplete as it can be controlled better, has more options, is being actively developed and supported, etc. However adding it would probably make more sense as part of reworking the Quick Edit/Bulk Edit functionality. Many things can be improved and coded better there.